The­or­ies 1-4

Guide writ­ten by LE★Baldy & Snaeky. Con­tri­bu­tions from the Amaz­ing Com­munity.

This guide is cur­rently un­der­go­ing change. Keep in mind, strategies may change.

The­ory ba­sics #

Pub­lic­a­tions are equi­val­ent to prestiges for \(f(t)\) so don’t be afraid to use them. However, the best pub­lic­a­tion mul­ti­pli­ers vary from the­ory to the­ory and will change over time. If you are close to a mul­ti­plier you want, turn off auto­buyer and let \(\rho\) in­crease without buy­ing up­grades for a faster short-term in­crease be­fore the pub­lic­a­tion (turn on after you pub­lish). This is known and ref­er­enced as “cruis­ing”. Total \(τ\), found in the equa­tion or at the top of the screen, is a mul­ti­plic­at­ive com­bin­a­tion of all \(τ\) from each the­ory.

Don’t be afraid to skip get­ting all mile­stones to work on the next or a bet­ter the­ory.

Note: If you see # → [# → # → #] → # in the mile­stone route of a the­ory, this is the sec­tion that has an act­ive strategy tied to it.

Gradu­ation rout­ing #

Re­mem­ber to fol­low our rout­ing ad­vice from In­tro­duc­tion to Gradu­ation.

The­ory 1 (20σ / 5k) #

T1 Over­view #

In math­em­at­ics, a re­cur­rence re­la­tion is an equa­tion that re­lies on an ini­tial term and a pre­vi­ous term to change. We start with the cur­rent tick’s term, \(ρ_{n}\), and a con­stant add-on to ob­tain the value of the next tick, \(ρ_{n+1}\). This gives us an equa­tion equi­val­ent to \(ρ=at+con­stant\), with a chan­ging value \(a\) and a con­stant that is the ini­tial value of 1. Later when we add the \(c_{3}ρ_{n-1}^{0.2}\) term, this is now say­ing that we are now adding each tick the value of \(ρ\) from the pre­vi­ous tick ago with a con­stant \(c_{3}\) put to the power of \(0.2\). This is the same with the next term \(c_{4}ρ_{n-2}^{0.3}\), with the value of \(ρ\) two ticks ago and a mul­ti­plier \(c_4\) put to the power \(0.3\). When we mul­tiply the \(c_1c_2\) term by the term \(1+ln(ρ)/​100\) chan­ging the con­stant ad­di­tion to be­ing based on the value of \(ρ\) from the pre­vi­ous tick with the value of \(1+ln(ρ)/​100\). The fi­nal mile­stone up­grade raises the ex­po­nent of \(c_1\) from \(1.00\) to \(1.05\) to \(1.10\) to \(1.15\).

This the­ory also has its ad­jus­ted tick­speed cal­cu­lated by \(q_{1}*q_{2}\). This lengthens the nor­mal tick length of \(0.1/​sec\) to that value which speeds up the the­ory.

T1 for­mula #

Ini­tial

\[ρ_{n+1} = ρ_n + c_1c_2\]

First mile­stone

\[ρ_{n+1} = ρ_n + c_1c_2 + c_3ρ_{n-1}^{0.2}\]

Second mile­stone

\[ρ_{n+1} = ρ_n + c_1c_2 + c_3ρ_{n-1}^{0.2} + c_4ρ_{n-2}^{0.3}\]

Third mile­stone

\[ρ_{n+1} = ρ_n + c_1c_2 \left( 1+\frac{ln(ρ_n)}{100} \right) \\\ + c_3ρ_{n-1}^{0.2} + c_4ρ_{n-2}^{0.3}\]

Fourth to Sixth mile­stone

\[ρ_{n+1} = ρ_n + c_1^{1.15}c_2 \left( 1+\frac{ln(ρ_n)}{100} \right) \\\ + c_3ρ_{n-1}^{0.2} + c_4ρ_{n-2}^{0.3}\]

T1 strategy #

The pub­lic­a­tion mul­ti­plier has no op­timal fit, as it fluc­tu­ates a lot, but here is known: 4-6 to start; 3-4 between 1e100 and 1e150; the pub­lic­a­tion mul­ti­plier os­cil­lates between 2.5 and 5 past e150. Once you get your second mile­stone, you can turn off \(c_1\) and \(c_2\) un­til e150 act­ive strat.

The act­ive strat fol­lows but only works when you have all mile­stones past e150. T1 is the only the­ory where the re­cent value of \(ρ\) in­flu­ences the rate of change of \(ρ\) there­fore buy­ing a vari­able as soon as you can af­ford it will slow your pro­gress. Lategame, buy­ing up­grades im­me­di­ately will slow you more than the be­ne­fit of the up­grade be­cause \(c_3\) & \(c_4\) dom­in­ate. If the next level costs \(10ρ\) and you have \(11ρ\), buy­ing that level will re­duce \(ρ_{n+1}\) to \(1\). This re­duces your \(ρ_{n+1}\) by roughly a factor of \(10\). There are \(3\) terms that in­flu­ence the rate of change of \(ρ\), and all are af­fected by the pre­vi­ous state of \(ρ\). The act­ive strategy around this is known as T1Ra­tio. The val­ues in the chart found here are to be only used when you are past \(e150 τ\) and max mile­stones. They rep­res­ent how to pur­chase each vari­able based on the state of the the­ory at the time of pur­chase.

Note: If you are not do­ing the act­ive strat, then simply turn off \(c_1\) and \(c_2\) after mile­stone 2 (e50τ) and auto­buy rest un­til ee6k.

The video be­low is only good for early tau between ee150 and ee250.

T1 mile­stone route #

The­ory 2 (25σ / 6k) #

T2 Over­view #

This second the­ory is fo­cus­ing on de­riv­at­ives. De­riv­at­ives in math­em­at­ics are the rate of change of the func­tion they are the de­riv­at­ive of. For the case of \(q_1\) and \(q_2\), \(q_2\) is the de­riv­at­ive of \(q_1\). This fol­lows the power rule for de­riv­at­ives:

\[q=a*t^n ↔ q’=n*a*t^{n-1}\]

In sim­pler terms, it works sim­ilar to how \(x_i\) up­grades work for \(f(t)\) equa­tion with con­tinu­ous ad­di­tion of the pre­vi­ous \(term*dt\) to the next \(x_{i+1}\) term, but with con­tinu­ous ad­di­tion of \(q_i*dt\) to the term above \(q_{i-1}\). These two val­ues of \(r_1\) and \(q_1\) are mul­ti­plied to pro­duce the de­riv­at­ive of \(ρ(t)\), shown by the new­ton de­riv­at­ive form \(\dot{ρ}\). This would give the equa­tion of \(ρ\) to be \(ρ(t+dt)=ρ+\dot{ρ}*dt\). The other mile­stones be­sides more \(q\) and \(r\) de­riv­at­ives in­crease the ex­po­nent of \(q\) and \(r\) re­spect­ively. The reason why \(q\) and \(r\) de­riv­at­ives are more power­ful long-term than the ex­po­nents is that they take time to build up and even­tu­ally over­take and keep in­creas­ing \(q_1\) and \(r_1\) while the ex­po­nents have a never-chan­ging boost.

T2 for­mula #

Ini­tial

\[\dot{q_n}=q_{n+1}*dt\] for n=1

\[\dot{r_k}=r_{k+1}*dt\] for k=1

\[\dot{ρ}=q_1r_1\]

First and Second mile­stones

\[\dot{q_n}=q_{n+1}*dt\] for n=1, 2, 3

\[\dot{r_k}=r_{k+1}*dt\] for k=1

\[\dot{ρ}=q_1r_1\]

Third and Fourth mile­stones

\[\dot{q_n}=q_{n+1}*dt\] for n=1, 2, 3

\[\dot{r_k}=r_{k+1}*dt\] for k=1, 2, 3

\[\dot{ρ}=q_1r_1\]

Fifth to Sev­enth mile­stones

\[\dot{q_n}=q_{n+1}*dt\] for n=1, 2, 3

\[\dot{r_k}=r_{k+1}*dt\] for k=1, 2, 3

\[\dot{ρ}=q_1^{1.15}r_1\]

Eight to Tenth mile­stones

\[\dot{q_n}=q_{n+1}*dt\] for n=1, 2, 3

\[\dot{r_k}=r_{k+1}*dt\] for k=1, 2, 3

\[\dot{ρ}=q_1^{1.15}r_1^{1.15}\]

T2 strategy #

The op­timal mul­ti­plier is pretty high and is not known be­fore \(e30\). The mul­ti­pli­ers for act­ive play we know at the mo­ment are:

• \(e25\)-\(e100\) is \(1k\) to \(10k\);
  
• \(e100\)-\(e175\) \(10k\)-\(100k\).

Idle

For the idle strategy, you want to pri­or­it­ize your mile­stones on x/​x/​0/​0 with \(q_{3}\) and \(q_{4}\) be­ing more im­port­ant than \(r_{3}\) and \(r_{4}\). If you have more than 5 mile­stones, you will pri­or­it­ize \(q\) ex­po­nent over the \(r\) ex­po­nent. You will want to pub­lish at about a \(1000\) mul­ti­plier, but lar­ger mul­ti­pli­ers are fine.

Act­ive

The goal of the act­ive strategy is to grow \(q_1\) and \(r_1\) as much as pos­sible while be­ing able to take ad­vant­age of the ex­po­nent mile­stones. The act­ive for T2 is on a 50-second cycle: 10 seconds on 0/​0/​x/​x mile­stones and 40 sec on x/​x/​0/​0 mile­stones. You will start a pub­lic­a­tion on 0/​0/​x/​x as the cost of the x/​x/​0/​0 mile­stone up­grades are too large for you to get right away. When you can af­ford them, you will start the cycle. This is what you will do for the fol­low­ing num­ber of mile­stones:

Past \(e175\), the act­ive strat will be­come ex­po­nen­tially less ef­fect­ive. At \(e250\), you would start to idle T2 overnight only. Un­til you are \(1e350\)+ \(τ\) for the­ory 2, this is the best the­ory to run idle overnight.

When you get to The­ory 3 at ee7k, move on to push­ing The­ory 3 \(τ\) and run­ning T2 overnight. The above is simply as an op­tion if you rather not work on T3 now.

T2 mile­stone route #

The­ory 3 (30σ / 7k) #

T3 Over­view #

The basis of this the­ory and un­der­stand­ing how it works is based on mat­rix mul­ti­plic­a­tion. Be­low I have put a color-coded im­age to dis­play how mat­rix mul­ti­plic­a­tion works.

This gives the basis for why cer­tain up­grades are more power­ful than oth­ers. The ex­po­nents on \(b_1\), \(b_2\), and \(b_3\) are all dir­ectly af­fect­ing \(ρ_1\) pro­duc­tion which is used for \(\tau\). An ex­tra di­men­sion roughly gives \(50%\) more \(\tau\) pro­duc­tion as it adds an ex­tra term to the \(ρ_1\) pro­duc­tion.

T3 strategy #

The op­timal pub­lic­a­tion mul­ti­plier is about 2-3 without cruis­ing and 3-4 with cruis­ing. If you de­cide to play act­ively, there is a form of ex­po­nent swap­ping strat to be aware of. This is a dif­fi­cult strategy be­cause it re­quires you to no­tice when a cer­tain threshold hap­pens. It hap­pens when the fol­low­ing oc­curs:

\[c_{11}*b_{1}^{1.05\text{ or }1.1}<c_{12}*b_{2}^{1.05\text{ or }1.1}\]

When this hap­pens swap your ex­po­nents from \(b_1\) to \(b_2\) and you will get a little up­grade boost. It also al­lows for a slight push of \(ρ_2\) for up­grades to \(b_2\) and \(c_{12}\), but this is a lot less im­pact­ful and less no­tice­able. This strategy also works with \(b_3\) and \(c_{13}\) but is of­ten­times not as com­mon and good to note any­ways.

If you de­cide to buy manu­ally, the fo­cus areas are \(b_1\), \(b_2\), and \(b_3\) when e1 lower than \(c_{11}\), \(c_{12}\), and \(c_{13}\). These all dir­ectly boost the pro­duc­tion of \(ρ_1\) which is used for \(\tau\). After this, if do­ing the act­ive ex­po­nent swap­ping strat in the pre­vi­ous para­graph, the next fo­cus will be on \(c_{21}\), \(c_{22}\), and \(c_{23}\) as these boost \(b_2\) pro­duc­tion which is the more likely cause for the ex­po­nent swap to oc­cur. This leaves the \(c_{31}\), \(c_{32}\), and \(c_{33}\) up­grades as the last pri­or­ity. If you are not us­ing the ex­po­nent swap­ping strat in the pre­vi­ous para­graph, then all the re­main­ing up­grades are at equi­val­ent pri­or­ity.

At the end of any pub­lic­a­tion, around a 2-3 mul­ti­plier, you should turn off \(b_1\) and \(c_31\) as they cost \(ρ_1\). You will cruise un­til you get to a 3-4 mul­ti­plier. Pub­lish and turn back on \(ρ_1\) cost­ing vari­ables and re­peat.

Com­ment­ary

T3 mile­stone route #

The­ory 4 (35σ / 8k) #

T4 Over­view #

We start out with just one term of con­stants \(c_1c_2\) and a chan­ging term \(c_3q\) with \(q\) be­ing equal to \(q(t+dt)=q+\dot{q}*dt\) with \(dt=0.1\) for each tick. \(\dot{q}\) is equal to an in­verse equa­tion of \(\dot{q}=q_1q_2/(​1+q)\) with \(q\) be­ing the cur­rent value. The first 3 mile­stones we grab add more terms to the \(ρ\) equa­tion with \(c_4q_2\), \(c_5q_3\), and \(c_6q_4\). Next, we in­crease \(\dot{q}\) by a factor of \(2^x\) up to \(2^3\) or \(8\). Fi­nally, we in­crease the power of \(c_1\) from \(1.00\) to \(1.15\).

T4 for­mula #

Ini­tial

\[\dot{ρ}=c_1c_2+c_3q\]

\[\dot{q}=q_1q_2/(​1+q)\]

First mile­stone

\[\dot{ρ}=c_1c_2+c_3q+c_4q^2\]

\[\dot{q}=q_1q_2/(​1+q)\]

Second mile­stone

\[\dot{ρ}=c_1c_2+c_3q+c_4q^2+c_5q^3\]

\[\dot{q}=q_1q_2/(​1+q)\]

Third mile­stone

\[\dot{ρ}=c_1c_2+c_3q+c_4q^2+c_5q^3+c_6q^4\]

\[\dot{q}=q_1q_2/(​1+q)\]

Fourth to Sixth mile­stones

\[\dot{ρ}=c_1c_2+c_3q+c_4q^2+c_5q^3+c_6q^4\]

\[\dot{q}=2^3q_1q_2/(​1+q)\]

Sev­enth mile­stone

\[\dot{ρ}=c_1^{1.15}c_2+c_3q+c_4q^2+c_5q^3+c_6q^4\]

\[\dot{q}=2^3q_1q_2/(​1+q)\]

T4 strategy #

The op­timal pub­lic­a­tion mul­ti­plier is 4-6. Dur­ing pub­lic­a­tions, start with x/​1/​3, then you will switch to 3/​0/​x. This will be re­peated back and forth throughout the pub­lic­a­tion. If you de­cide to manu­ally buy and don’t have max mile­stones, fo­cus on \(q_1\) and \(q_2\). The next pri­or­ity is go­ing from the highest \(c_x\) up­grade down to \(c_1\). Each lower pri­or­ity should be bought \(e1\) cheaper than the pri­or­ity tier above. If you de­cide to manu­ally buy at max mile­stones, at the be­gin­ning of pub­lic­a­tions, buy \(c_1\), \(c_2\), \(c_3\), \(q_1\), and \(q_2\). Once you are within \(e1\)-\(e2\) of your pub­lic­a­tion mark, swap to only buy­ing \(c_3\), \(q_1\), and \(q_2\).

T4 mile­stone route #

The­ory tier list (Pre-9k+) #

Be­fore you reach 9k, these are the re­com­men­ded val­ues for each the­ory. You may not hit the val­ues and have a dif­fer­ent dis­tri­bu­tion, but work on get­ting these the­or­ies up to these val­ues later. This list is in or­der of pri­or­ity.